Tracking the Frequency Moments at All Times

TL;DR

This paper improves the understanding of tracking frequency moments in streaming algorithms, showing that fewer instances are needed for certain models and establishing tight bounds that match the standard approach's optimality.

Contribution

It provides tighter upper bounds for the number of instances needed in the tracking problem and proves matching lower bounds, demonstrating the optimality of the standard approach in various models.

Findings

Fewer instances needed for $F_p$ tracking in the cash register model.

Lower bounds match the upper bounds, confirming optimality.

Standard approach is essentially optimal in the turnstile model.

Abstract

The traditional requirement for a randomized streaming algorithm is just {\em one-shot}, i.e., algorithm should be correct (within the stated $\eps$-error bound) at the end of the stream. In this paper, we study the {\em tracking} problem, where the output should be correct at all times. The standard approach for solving the tracking problem is to run $O(\log m)$ independent instances of the one-shot algorithm and apply the union bound to all $m$ time instances. In this paper, we study if this standard approach can be improved, for the classical frequency moment problem. We show that for the $F_p$ problem for any $1 < p \le 2$, we actually only need $O(\log \log m + \log n)$ copies to achieve the tracking guarantee in the cash register model, where $n$ is the universe size. Meanwhile, we present a lower bound of $\Omega(\log m \log\log m)$ bits for all linear sketches achieving this guarantee. This shows that our upper bound is tight when $n=(\log m)^{O(1)}$. We also present an $\Omega(\log^2 m)$ lower bound in the turnstile model, showing that the standard approach by using the union bound is essentially optimal.